Question

A particle of mass $m$ is moving in a straight line. Starting at time $t=0$, a force $F=kt$ acts in the same direction on the moving particle during time interval $T$ so that its momentum changes from $p$ to $3p$. Here $k$ is a constant. The value of $T$ is :

• A. $2\sqrt{\frac{k}{p}}$
• B. $2\sqrt{\frac{p}{k}}$
• C. $\sqrt{\frac{2k}{p}}$
• D. $\sqrt{\frac{2p}{k}}$

Answer 1

The correct option is B $2\sqrt{\frac{p}{k}}$

From Newton's second law

$\frac{dp}{dt}=F=kt$

Integrating both sides we get,

${\int }_p^{3p}dp={\int }_0^{T}ktdt$

$\Rightarrow [p{]}_p^{3p}=k{\left[\frac{t^2}{2}\right]}_0^T$

$\Rightarrow 2p=\frac{k{T}^2}{2}$

$\Rightarrow T=2\sqrt{\frac{p}{k}}$

Hence, $\left(B\right)$ is the correct answer.